[racket] Math library kudos

Hi Joe,

2013/2/20 Joe Gilray <jgilray at gmail.com>:
> Racketeers,
>
> Thanks for putting together the fantastic math library.  It will be a
> wonderful resource.  Here are some quick impressions (after playing mostly
> with math/number-theory)
>
> 1) The functions passed all my tests and were very fast.  If you need even
> more speed you can keep a list of primes around and write functions to use
> that, but that should be rarely necessary

That's great to hear.

> 2) I have a couple of functions to donate if you want them:

Contributions to the library is very welcome. For example
it would be nice to have a better implementation of next-prime.

> 2a) Probablistic primality test:
>
> ; function that performs a Miller-Rabin probabalistic primality test k
> times, returns #t if n is probably prime
> ; algorithm from http://rosettacode.org/wiki/Miller-Rabin_primality_test,
> code adapted from Lisp example
> ; (module+ test (check-equal? (is-mr-prime? 1000000000000037 8) #t))
> (define (is-mr-prime? n k)
>   ; function that returns two values r and e such that number = divisor^e *
> r, and r is not divisible by divisor
>   (define (factor-out number divisor)
>     (do ([e 0 (add1 e)] [r number (/ r divisor)])
>       ((not (zero? (remainder r divisor))) (values r e))))
>
>   ; function that performs fast modular exponentiation by repeated squaring
>   (define (expt-mod base exponent modulus)
>     (let expt-mod-iter ([b base] [e exponent] [p 1])
>       (cond
>         [(zero? e) p]
>         [(even? e) (expt-mod-iter (modulo (* b b) modulus) (/ e 2) p)]
>         [else (expt-mod-iter b (sub1 e) (modulo (* b p) modulus))])))
>
>   ; function to return a random, exact number in the passed range
> (inclusive)
>   (define (shifted-rand lower upper)
>     (+ lower (random (add1 (- (modulo upper 4294967088) (modulo lower
> 4294967088))))))
>
>   (cond
>     [(= n 1) #f]
>     [(< n 4) #t]
>     [(even? n) #f]
>     [else
>      (let-values ([(d s) (factor-out (- n 1) 2)]) ; represent n-1 as 2^s-d
>        (let lp ([a (shifted-rand 2 (- n 2))] [cnt k])
>          (if (zero? cnt) #t
>              (let ([x (expt-mod a d n)])
>                (if (or (= x 1) (= x (sub1 n))) (lp (shifted-rand 2 (- n 2))
> (sub1 cnt))
>                    (let ctestlp ([r 1] [ctest (modulo (* x x) n)])
>                      (cond
>                        [(>= r s) #f]
>                        [(= ctest 1) #f]
>                        [(= ctest (sub1 n)) (lp (shifted-rand 2 (- n 2))
> (sub1 cnt))]
>                        [else (ctestlp (add1 r) (modulo (* ctest ctest)
> n))])))))))]))

As far as I can tell, this is the same algorithm as prime-strong-pseudo-single?
from number-theory.rkt. See line 134.

https://github.com/plt/racket/blob/master/collects/math/private/number-theory/number-theory.rkt

--
Jens Axel Søgaard